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Zagazig University
Faculty of Engineering
Electrical Power and Machines Department
ROBUST DECENTRALIZED CONTROLLER
DESIGN VIA AI TO ENHANCE POWER
SYSTEM DYNAMIC PERFORMANCE
Prepared by
EHAB SALIM ALI MOHAMMED SALAMA
M.Sc. & B.Sc. in Electrical Engineering, Faculty of
Engineering ,Zagazig University
A Thesis
Submitted to the Faculty of Engineering in Partial Fulfillment of the
Requirements For The Degree of Doctor of Philosophy (Ph.D.) in
Electrical Engineering
Supervised By:
Prof. Dr. M. E.Mandour Prof. Dr. Z. S. El-RazazProf. of Electrical power Prof. of Electrical power
Zagazig university Zagazig university
2006
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"
"
)113(
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Zagazig University
Faculty of Engineering
Electric Power and Machines Department
ROBUST DECENTRALIZED CONTROLLER DESIGN
VIA AI TO ENHANCE POWER SYSTEM DYNAMIC
PERFORMANCE
Prepared by
EHAB SALIM ALI MOHAMMED SALAMA
A Thesis Submitted in Partial Fulfillment of the Requirements for the
Degree of Ph.D. in Electrical Engineering.
Approved by the examining committee
Prof. Dr. M. M. El-Metwally
Prof. Dr. F. M. A. Bendary
Prof. Dr. M. E. Mandour
Prof. Dr. Z. S. El-Razaz
2006
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Acknowledgement
First of all, I would like to express all thanks to God and I look forward for hisassistance.
My deep appreciation goes to Prof Dr. M. E. Mandour and Prof. Dr. Z. S. El-
Razaz for their valuable guidance, suggestions, continuous encouragement during the
progress of this work Moreover, my thanks go to the staff of the electrical power and
machines department.
My special thanks are for my mother for her prayers for me, which were a greathelp to me to complete this work. And I am really grateful to my wife for her great
help and co-operation.
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ABSTRACT
Power systems are modeled as large-scale systems composed of a set of small-
interconnected subsystems. It is generally impossible to incorporate many feed back
loops into the controller design for large scale interconnected systems and is also too
costly even if they can be implemented. These motivate the development of
decentralized control theory where each subsystem is controlled independently on its
local available information.
On the other hand, the operating conditions of power systems are always varying to
satisfy different load demands. Control systems are therefore required to have the
ability to damp the system oscillations that might threaten the system stability as the
load demand increases. However, as power systems are large-scale nonlinear systems
in nature, the applications of conventional power system stabilizer (PSS) are limited.
There is thus a need for controllers, which are robust to changes in the system
operating condition. Robust controllers based on control theory are particularly
suited for this purpose.
H
This thesis proposes two robust decentralized controllers for multimachine power
system instead of using a complex centralized controller. The first one is based on
theory, and results in high order controller. The second controller is a
proportional integral (PI) type, and is tuned by a novel robust performance as the first
one, but it is more appealing from an implementation point of view. In more detail,
the second control design is first cast into the robust
H
H control design in terms of
linear matrix inequalities (LMI) in order to obtain robustness against system operating
conditions. An additional constraint is that the structure of the controller is predefined
as a PI type, which is ideally practical for industry. In order to obtain the optimal
controller parameters with regards to the
H and controller structure constraints,
genetic algorithms (GAs), a powerful probabilistic search technique is used to find the
control parameters of the PI controller.
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To treat the problem of possible adverse interaction between multiple decentralized
controllers, three global control strategies are introduced in this thesis. The first,
which is a multi-input multi-output (MIMO) centralized with effective
communicated information. The second is the reduced centralized based on the
balanced truncation method. While the third is based on two level PSS. The
simulation results show that the proposed controllers ensure adequate damping for
widely varying system-operating conditions.
H
H
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Abbreviations
AI Artificial Intelligence
ANN Artificial Neural Network
ASVC Advanced Static Var Compensator
FACTS Flexible AC Transmission System
GA Genetic Algorithms
LFC Load Frequency Control
OC Optimal Control
PI Proportional Integral Controller
PID Proportional Integral Differential Controller
PSO Particle Swarm Optimization
PSS Power System Stabilizer
LMI Linear Matrix Inequalities
LQ Linear Quadratic
SA Simulated Annealing
SAPSS Simulated Annealing Based Power System Stabilizer
SMIB Single Machine Infinite Bus
SSV Structure Singular Values
STATCOM Static Compensator
SVC Static Var Compensator
TCSC Thyristor Controlled Series Capacitor
TS Tabu Search
UPFC Unified Power Flow Controller
Vref Reference Voltage
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List of Symbols
Angular speed.
The deviation from nominal values.
V Infinite bus voltage.
tV Generator terminal voltage.
ejX
eR + Transmission line resistance and inductance.
fdE Exciter voltage.
aT
aK , Gain and time constant of the excitation system.
fT
fK ,
Gain and time constant of the field system.
mT Mechanical input torque.
eT Electrical torque.
dI ,
qI d-and q-axis terminal current respectively.
Ido, Iqo d -and q-axis nominal current respectively.
j Inertia coefficient, H
j2= .
VqoVdo, The nominal voltage in d and q axes in p.u.
qE' Internal voltage behind in p.u.d
X'
qoE Q axis voltage.
Torque angle in rad.
do` Time constant of excitation system in sec.
B Rated angular speed.
dX ,
qX d-and q-axis reactance of the generator respectively.
'dX The d- axis transient reactance of the generator.
H Inertia constant.
DQ A state weighting matrix.
DR A control weighting matrix.
K Controller.
DCBA ,,, The state space equation.
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rD
rC
rB
rA ,,, The state space equation of reduced-order model.
kD
kCB
kA k ,,, The state space equation of controller.
clD
clC
clB
clA ,,, The state space equation of closed loop system.
CA = coupling block matrix A.ijA
DA = decoupling block matrix A.iiA
C
B = coupling block matrix B.ijB
D
B = decoupling block matrix B.iiB
CC = coupling block matrix C.ijC
DC = decoupling block matrix C.iiC
lu Local control signal.
gu
Global control signal.
tx )( Denote the state vector.
tW )( The vector of input disturbance.
tu )( The vector of control input.
ty )( The vector of measured variables.
tZ )( The vector of error signals.
)(sZWT The closed loop transfer matrix from the disturbance W to the
regulator output Z.
(gopt) The norm of the transfer function .)(sZWT
ST ,
dT The change in the synchronizing and damping torque
component respectively.
s
K ,
d
K The synchronizing and damping torque coefficient respectively.
Kp, Ki PI controller gains.
wT Washout time constant.
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Contents
Contents
page
Acknowledgment
Abstract
Abbreviations
List of Symbols
Contents
List of Figures
List of Tables
I
II
IV
V
VII
X
XIV
Chapter 1 Introduction
1.1 Power System and Robust Control Technique. 1
1.2 Power System Modes of Oscillation. 2
1.3 Power System Stabilizers. 3
1.4 Decentralized Control. 4
1.5 Conflict Between Centralized and Decentralized
Controller in PSS Design.
6
1.6 Thesis Objectives. 7
1.7 Outline of The Thesis. 8
Chapter 2 Review of Literature
2.1 Introduction. 11
2.2 Previous Work. 11
2.3 Contributions of This Thesis. 24
Chapter 3 Modeling of Power Systems
3.1 Introduction. 26
3.2 System Equations. 26
3.3 Block Diagram Simulation. 28
3.4 State Space Formulation. 32
3.5 Formulation of The System Model. 35
3.6 System Under Study. 393.7 State Space Equations. 42
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Contents
Chapter 4 Centralized and Decentralized Controllers
4.1 Introduction. 46
4.2 Large Scale Controllers. 46
4.3 System Representation. 48
4.3.1 State and Output Feedback 49
4.4 Design of The Optimal Decentralized Controller. 50
4.5 Design of The Sub optimal Decentralized Controller. 52
4.6
H Robust Controller. 53
4.6.1 Linear Matrix Inequality (LMI). 56
4.6.2 Robust
H Control Design Via LMI. 58
4.7 Robust Controller Design Via Reduced Order Model. 60
4.8 Limitations and Shortcomes of Previous Mentioned
Controllers.
62
4.9 Proposed Robust Control Design Via GALMI. 63
4.9.1 An Overview of Genetic Algorithms. 64
4.9.2 An Overview of Particle Swarm Optimization. 66
4.10 Global Controller Design. 68
4.11 Proposed Two Level PSS Controller Design. 71
Chapter 5 Robust Controller Design for Single Machine Infinite
Bus
5.1 Introduction. 73
5.2 Dynamic Model of SMIB. 73
5.3 Implementation of
H Controller. 76
5.4 Balanced Truncation
H Controller Implementation. 78
5.5 Implementation of GALMI Controller. 85
5.6 Recommendation for Multimachine System 93
Chapter 6 Robust Decentralized Controller Design for
Multimachine System
6.1 Introduction. 94
6.2 Evaluation of Multimachine System. 96
6.2.1 System Modes Classification. 96
6.2.2 Block Diagram Simulation of Multimachine
System.
98
6.2.3 Response of The System Without Controller. 100
6.2.4 Effect of Loading on System Dynamic. 101
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Contents
6.3 Implementation of Optimal Controller. 102
6.4 Implementation of Sub optimal Controller. 107
6.5 Fixed Modes Problem Formulation. 111
6.6 Centralized and Decentralized Controller Design Via.
H
113
6.7 Dynamic Model of Multimachine System 120
6.8 Simulation and Evaluation of GALMI 122
6.9 Implementationof Global Controller 125
6.10 Implementation of The Proposed Two Level Controller 131
6.11 System Performance With Two Subsequence
Disturbances
135
Chapter 7 Conclusions and Recommendations
7.1 Conclusions of This Thesis. 137
7.2 Recommendations for Future Work. 138
References 140
Appendix A 147
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List of Figures
List of Figures
Chapter 1 IntroductionFigure (1.1) Block diagram of PSS. 3
Figure (1.2) Schematic diagram of centralized controller. 5
Figure (1.3) Schematic diagram of decentralized controller. 6
Chapter 2 Review of literature
Chapter 3 Modeling of Power Systems
Figure (3.1) Machine-infinite bus. 26
Figure (3.2) The block diagram of a single machine adopted to
be used in Simulink Toolbox.
31
Figure (3.3) Conversion from machine axes to common frame
axes.
35
Figure (3.4) Block diagram representation of a single machine
connected to the network.
39
Figure (3.5) System under study. 40
Figure (3.6) Matrix Q1. 42
Figure (3.7) Matrix Q2. 43
Figure (3.8) Matrix A0. 44
Figure (3.9) Matrix Gxs. 45
Chapter 4 Decentralized and Centralized Controllers
Figure (4.1) Flow chart of the controllers 47
Figure (4.2) Generalized block diagram of
H . 54
Figure (4.3) Flow chart of the GA optimization 65
Figure (4.4) Flow chart of the PSO optimization 67
Figure (4.5) Flow chart of the local and global controller 70
Figure (4.6) Proposed two level PSS design 72
Chapter 5 Robust Controller Design for Single Machine
Infinite Bus
Figure (5.1) Response of for 0.1 p.u step in Vref for testedoperating point
77
Figure (5.2) Response of for 0.1 p.u step in Vref for tested
operating point.
78
Figure (5.3) Response of for 0.1 p.u step in Vref for first
operating point
80
Figure (5.4) Response of for 0.1 p.u step in Vref for first
operating point
80
Figure (5.5) Change in control signal for 0.1 p.u step in Vref 81
Figure (5.6) Bode plot of the transfer function for first
operating point.
82
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List of Figures
Figure (5.7) Response of for 0.1 p.u step in Vref for
second operating point.
83
Figure (5.8) Response of for 0.1 p.u step in Vref for
second operating point.
83
Figure (5.9) Response of for 0.1 p.u step in Tm for second
operating point.
84
Figure (5.10) Response of for 0.1 p.u step in Vref for first
operating condition
85
Figure (5.11) change in control signal for 0.1 p.u step in Vref 86
Figure (5.12) Variations of objective function 87
Figure (5.13) Variations of Kp 87
Figure (5.14) Variations of Ki 88
Figure (5.15) Response of for different values of (gopt)
for P=1.0, Q=0.4
90
Figure (5.16) Response of for 0.1 p.u step in Vref for
second operating condition.
91
Figure (5.17) Response of for 0.1 p.u step in Tm for second
operating condition.
92
Figure (5.18) Bode plot of the transfer function. 92
Chapter 6 Robust Decentralized Controller Design For
Multimachine System
Figure (6.1) System under study. 96Figure (6.2) The block diagram of the system under study. 99
Figure (6.3) Response of12
to 0.1 p.u step in Vref. 100
Figure (6.4) Response of to 0.1 p.u step in Vref.12
100
Figure (6.5) Open loop poles (mechanical modes) for the 9
bus, 3 machine system.
102
Figure (6.6) Response of13
w for 0.1 step in Vref of Gen. (1) 106
Figure (6.7) Response of23
w for 0.1 step in Vref of Gen. (1) 106
Figure (6.8) Response of12
for 0.1 step in Vref of Gen. (1) 110
Figure (6.9) Response of13
w for 0.1 step in Vref of Gen. (1) 110
Figure (6.10a) Schematic of centralized output feedback
controller.
114
Figure (6.10b) Schematic of decentralized output feedback
controller.
114
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List of Figures
Figure (6.11) The response of12
w for light load condition. 115
Figure (6.12) The response of23
w for light load condition. 115
Figure (6.13) The response of
12
w fornormal load condition. 116
Figure (6.14) The response of23
w fornormal load condition. 117
Figure (6.15) The response of12
w for heavy load condition. 119
Figure (6.16) The response of23
w for heavy load condition. 120
Figure (6.17) Response of12
for light load condition with
three PI local decentralized controllers.
123
Figure (6.18) Response of 12 for normal load condition with
three PI local decentralized controllers.
124
Figure (6.19) Response of23
w for heavy load condition with
three PI local decentralized controllers.
125
Figure (6.20) Response of13
for light load condition due to
different robust global controllers.
126
Figure (6.21) Response of13
w for light load condition due to
different robust global controllers.
127
Figure (6.22) Response of13
for normal load condition due
to different robust global controller.
128
Figure (6.23) Response of12
w for normal load condition due
to different robust global controllers.
128
Figure (6.24) Response of13
for heavy load condition due to
different robust global controllers.
130
Figure (6.25) Response of13
w for heavy load condition due to
different robust global controllers.
131
Figure (6.26) Region in the left hand side of a vertical line 131
Figure (6.27) Comparison of13
response for normal load
condition with different robust damping global
controllers.
132
Figure (6.28) Comparison of12w response for normal load
condition with different robust damping globalcontrollers.
133
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List of Figures
Figure (6.29) Comparison of12
w response for heavy load
condition with different robust damping global
controllers.
134
Figure (6.30) Comparison of13
w response for heavy
condition with different robust damping global
controllers.
134
Figure (6.31) Response of12w under two subsequence
disturbances.
135
Figure (6.32) Response of13
w under two subsequence
disturbances.
136
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List of Tables
List of Tables
Chapter 1 Introduction
Chapter 2 Review of literature
Chapter 3 Modeling of Power Systems
Table (3.1) Bus data for the base case (Load flow) on the 100
MVA Base.
40
Table (3.2) Transmission lines and transformer data all values
are in p.u. on 100 MVA base.
41
Table (3.3) Generator data: Reactance values are in pu on a
100-MVA base.
41
Chapter 4 Decentralized and Centralized Controllers
Chapter 5 Robust Controller Design for Single Machine
Infinite Bus
Table (5.1) Eigenvalues of closed loop system with different
controllers.
79
Table (5.2) Comparison between three controllers for first
operating point
89
Table (5.3) Comparison between GA and PSO. 89
Chapter 6 Robust Decentralized Controller Design For
Multimachine System
Table (6.1) A part of the participation matrix corresponding to
the mechanical modes.
97
Table (6.2) System modes after classification. 97
Table (6.3) The eigenvalues, and frequencies associated with
the rotor oscillation modes of the system.
98
Table (6.4) Loading conditions for the 9 bus, 3 machine
system ( in p.u).
101
Table (6.5) Open loop eigenvalues of the rotor oscillation
modes of the system.
102
Table (6.6) System modes with optimal decentralized
controller and centralized one.
105
Table (6.7) The eigenvalues, and damping ratios associated
with the rotor oscillation modes of the system for
both controllers.
105
Table (6.8) System modes with sub optimal decentralized and
centralized controller.
109
Table (6.9) The eigenvalues, and damping ratios associated
with the rotor oscillation modes of the system for
both controllers.
109
Table (6.10) System modes with variable controller. 112
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List of Tables
Table (6.11) Eigenvalues of closed loop system with
centralized and decentralized controllers for 1.0
p.u (normal load).
118
Table (6.12) Eigenvalues of closed loop system for different
operating conditions with three GALMI
controllers.
123
Table (6.13) Eigenvalues of closed loop system with global and
reduced global controller for 1.00 p.u (normal
load).
129
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Chapter 1 Introduction
Chapter 1
Introduction
1.1 Power System and Robust Control Technique
Power systems are usually large nonlinear systems, which are often subject to low
frequency oscillations when working under some adverse loading conditions. The
oscillation may sustain and grow to cause system separation if no adequate damping
is available. To enhance system damping, the generators are equipped with power
system stabilizers (PSSs) that provide supplementary feedback stabilizing signals in
the excitation systems. PSSs enhance the power system stability limit by improving
the system damping of low frequency oscillations associated with the
electromechanical modes. Many approaches are available for PSS design, most of
which are based either on classical control methods or on intelligent control strategies.
Power systems continually undergo changes in the operating condition due to changes
in the loads, generation, and in the transmission network resulting in accompanying
changes in the system dynamics. A well-designed stabilizer has to perform
satisfactorily in the presence of such variations in the system. In other words, the
stabilizer should be robust to changes in the system over its entire operating range.
The nonlinear differential equations, which simulate the behavior of a power system,
can be linearized at a particular operating point to obtain a linear model, which
represents the small signal oscillatory response of the power system. Any variation in
the operating condition of the system may cause a variation in the system model. For
a different variation in the operating conditions of a particular system a set of a linear
models, each corresponding to one particular operating condition may be generated.
Since, at any given instant, the actual plant could correspond to any model in this set,
a robust controller would have to impart adequate damping to each one of these entire
sets of linear models.
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Chapter 1 Introduction
Robust control technique has been applied to power system controller design since
1980s. The main advantage of this technique is that it presents a natural tool for
successfully modeling plant variations. Several studies, which will be mentioned in
previous work, have been devoted to the design of power system controllers for PSS
and/or FACTS devices using
H . In these studies, many classical control
objectives such as disturbance attenuation, robust stabilization of power systems are
expressed in terms of performance and tackled by
H
Hsynthesis techniques.
The control problem is to find a controller that minimizes
H where
= )(sZWT and represents the norm of the transfer function of the output (Z)
to the disturbance (W). In other words, minimize the energy of the output signals (Z)
for a given set of exogenous signals (W). All these studies produce a controller, which
is robust. These controllers provide added damping to the system under a wide
range of operating conditions.
1.2 Power System Modes of Oscillation
An electrical power system consists of many individual elements connected together
to form a large, complex system capable of generating, transmitting and distributing
electrical energy over a large geographical area. Due to these interconnections of
elements, a large variety of dynamic interactions are possible to done which may
affect on the system.
The stability problem involves the study of the electromechanical oscillations inherent
in power systems [1]. Power systems exhibit various modes of oscillation due to
interactions among system components. Power systems usually have two distinct
forms of oscillations.
1- Local modes are associated with the swinging of units at a generating station
with respect to the rest of the power system. The term local is used because the
oscillations are localized at one station or a small part of the power system.
Typical local-mode frequency range from 0.8-2.0 Hz.
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Chapter 1 Introduction
2- Inter-area modes are associated with the swinging of many machines in one
part of the system against machines in other parts. They are caused by two or
more groups of closely coupled machines being interconnected by weak ties.
Typically have a frequency in the range from 0.1-0.8 Hz.
Undamped oscillations once started often grow in magnitude over the span of many
seconds. Sustained oscillations in the power system are undesirable for many reasons.
They can lead to fatigue of machine shafts, cause excessive wear of mechanical
actuators of machine controllers and also make system operation more difficult. It is
therefore desirable that oscillations are well damped. So PSS is necessary to provide
appropriate damping of undesirable oscillations caused by disturbances.
1.3 Power System Stabilizers
The basic function of a power system stabilizer is to extend the stability limits by
adding damping to generator rotor oscillations by controlling its excitation using
auxiliary stabilizing signal(s). To provide damping, the stabilizer must produce a
component of electric torque, which is in phase with rotor speed deviations. The
oscillations of concern typically occur in the frequency range of approximately 0.1 to
2.0 Hz, and insufficient damping of these oscillations may limit the ability to transmit
the power. The block diagram used in industry is shown in Figure (1.1). It consists of
a washout circuit, phase compensator (lead-lag circuit), stabilizer gain and limiter [2].
Figure (1.1) Block diagram of PSS
KSTAB
Stabilizer gain Washout Lead-Lagumax
umin
usTw
1+sTw
1+sT11+sT2
1+sT31+sT4
Lead-Lag
The phase compensation block provides the appropriate phase lead characteristic to
compensate for the phase lag between the exciter input and generator electrical
torque. The required phase lead can be obtained by choosing the values of time
constants .41,....,TT
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Chapter 1 Introduction
The signal washout block serves as a high pass filter, with the time constant high
enough to allow signals associated with oscillations in speed to pass as it. Without it,
steady changes in speed would modify the terminal voltage. It allows the PSS to
respond only for a change in the speed. From the viewpoint of the washout function,
the value of is not critical and may be in the range of 1 to 20 seconds.
WT
WT
The stabilizer gain determines the amount of damping introduced by PSS.
Ideally the gain should be set at a value corresponding to maximum damping.
However, in practice the gain is set to a value that results in satisfactory damping of
the critical system modes without compromising the stability of other modes.
STABK
In order to restrict the level of generator terminal voltage fluctuation during transient
conditions, limits are imposed on PSS outputs.
1.4 Decentralized Control
The complexity and high performance requirements of present day industrial
processes place increasing demands on control technology. The orthodox concept of
driving a large system by a central controller has become unattractive for either
economic or reliability reason. New emerging notions are subsystems,
interconnections, parallel processing, and information constraints, to mention a few.
In complex system, where databases are developed around the plants with distributed
sources of data, a need for fast control action in response to local inputs and
perturbations dictates the use of distributed (that is, decentralized ) information and
control structures.
The accumulated experience in controlling complex system suggests three basic
reasons for using decentralized control structures:
1- dimensionality ,
2- information structure constraints, and
3- uncertainty.
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Chapter 1 Introduction
Because the amount of computation required to analyze and control a large-scale
system grows faster than its size, it is beneficial to decompose the system into
subsystems, and design controls for each subsystem independently based on local
subsystem dynamics and its interconnections. In this way, special structure features
of a system can be used to devise feasible and efficient decentralized strategies for
solving large control problems previously impractical to solve by one shot
centralized methods.
A restriction on what and where the information is delivered in a system is a standard
feature of interconnected systems. For example, the standard automatic generation
control in power system is decentralized because of the cost of excessive information
requirements imposed by a centralized control strategy over distant geographic areas.
The structure constraints on information make the centralized methods for control and
estimation design difficult to apply, even to systems with small dimensions.
It is a common assumption that neither the internal nor the external nature of complex
systems can be known precisly in deterministic or stochastic terms. Decentralized
control strategies are inherently robust with respect to a wide variety of structure and
unstructured perturbations in complex systems[3].
Figures (1.2-1.3) represent a schematic diagram of centralized and decentralized
controller respectively.
K
G-1 G0 G1 G2
Figure (1.2) Schematic diagram of centralized controller
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Chapter 1 Introduction
G-1 G0 G1 G2
K -1 K 0 K 1 K 2
Figure (1.3) Schematic diagram of decentralized controller
1.5 Conflict Between Centralized and Decentralized Controller in
PSS Design
Two basic approches are avaliable for designing PSS. The first approach is to use a
multi-input multi-output centralized controller which would require a significant
amount of system wide communication. With this approach, the controller success
heavily depends on the communication which can be a serious disadvantage. Also a
controller failure might paralyze the whole network. So centralization is undesirable.
The second approach is the decentralized controller schemes, which have a number of
advantages. From which its operation requires a local signal so it is easy to design as
a hardware. So a failure of one controller has no detrimental effect on the
performance of the other controllers. Also the dependence on communication between
control stations is greatly reduced.
In the centralized PSS, the control signal to a machine is a function of the outputs for
all the machines. This affects on the gain matrix of centralized PSS for multimachine
power system by making it full. So a transmitted signals among the generating units is
needed. For this the centralized controller system is complex to design and
implement.
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Chapter 1 Introduction
In the decentralized PSS, the control signal for each machine should be a function of
its output only. So the gain matrix should have a zeros for all offdiagonal terms. For
this the decentralized control scheme is preferable.
1.6 Thesis Objectives
For large power systems comprising many interconnected machines, the PSS
parameter tuning is a complex task due to the presence of several poorly damped
modes of oscillation. The problem is further complicated by continuous variation in
power system operating conditions. To meet modern power system requirements,
controllers have to guarantee stability and robustness over a wide range of system
operating conditions. Thus the robustness is one of the major issues in power system
controllers design. So the recently developed
H synthesis is the way to handle
these requirements.
The main objectives of the thesis:
1- Design a robust controller for single machine infinite bus (SMIB)
a) Design a robust controller based on the
H .
b) Design a reduced controller based on balanced truncation method.
c) Design a proportional integral (PI) controller using the genetic algorithms
(GAs), which is well known as the new generation of the artificial
intelligence (AI). The parameters of the PI controller are tuned to mimic
the robust performance of the
H optimal one designed in (a). In other
word, the parameters of the PI controller have to obtain the same as that
of . More specifically, GAs is used to obtain the control parameters
of the PI controller subject to the
H
H constraints in terms of linear matrix
inequalities (LMI). Hence, this control design is called GALMI.
d) Another optimization tool, which is particle swarm optimization (PSO) is
used to tune PI controller and ensure a best solution.
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Chapter 1 Introduction
2- Design a robust controller for multimachine system
a) Design a centralized and decentralized robust controller based on the
optimal control method for PSS.
H
b) Design a GALMI for each area instead of the robust local decentralized
controller.
H
3- Design global controller to coordinate between the decentralized controllers
and ensure stability of the interconnected system.
a) Design a global controller based on multi-input multi-output (MIMO)
centralized controller.
H
b) Design a global controller based on reduced MIMO centralized
controller via balanced truncation method.
H
c) Design a global controller based on two level PSS. The parameters of PI
controller are selected to shift the undamped mechanical modes of
oscillation to the left hand side of vertical line in the complex s- plane by
GA.
1.7 Outline of The Thesis
The general description of the thesis is as follow:
Chapter 1: Discuss power system modes of oscillation and the classical structure of
PSS in brief. Also, conflict between centralized and decentralized controller in PSS
design are investigated. Moreover, introduces the reader to the thesis objectives and
outlines of their chapters.
Chapter 2: Performs a survey of the previous work, which discusses the relevant
work in the area of tuning PSS and robust control.
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Chapter 1 Introduction
Chapter 3: Presents the steady state and dynamic models of a SMIB and
multimachine power system. The mathematical formulations have been manipulated
in details. The machine models have been formulated in state-space form.
Chapter 4: Illustrates the design of centralized and decentralized controllers using
optimal and sub-optimal method for multimachine power system. Moreover, three
robust controllers are proposed. The first is based on
H theory, and results in a
high number of states, which represents the order of controller. The second controller
is a reduced order controller based on balanced truncation. The third is the PI
controller, has a simple structure, which is more appealing from an implementation
point of view, and it is tuned by GAs to achieve the same robust performance as thefirst one. More specifically, GAs optimization is used to tune the control parameters
of the PI controller subject to the
H constraints in terms of LMI. Hence, the third
control design is called GALMI. Particle swarm optimization (PSO), which is another
optimization technique, is used to tune the PI controller and ensure the best solution.
The previous controllers are further extended for designing centralized and
decentralized controller for multimachine power system based on . Moreover,
three global control strategies are introduced in this thesis to coordinate between the
decentralized controllers and ensure stability of the interconnected system. The first
utilizes a MIMO centralized
H
H controller system. The second is based on reduced
MIMO centralized controller via balanced truncation method. The third strategy
utilizes a two level PSS based on PI controller.
H
Chapter 5: The implementation and application of the three robust controllers are
illustrated for SMIB system. The first one is based on
H theory, and results in a
high order controller. The second controller is the reduced one based on balanced
truncation. The third controller design is based on GAs and is called GALMI. PSO is
used instead of GAs to redesign the third controller. The performance of the
implemented controllers is obtained for different operating conditions.
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Chapter 1 Introduction
Chapter 6: In this chapter, the implementation of the decentralized and centralized
controller based on optimal control theory and a comparison between them is
presented. Also the problem of fixed mode is presented. Moreover, a novel robust
decentralized controller with a simple structure is introduced based on the
optimal control method for PSS in multimachine area. The parameters of the PI
controller are tuned to mimic the robust performance of the decentralized one.
GALMI is used to obtain the control parameters of the PI controller. A global
controller is developed by reduced centralized controller based on minimum
communicated information to coordinate the local decentralized controllers. The
reduced controller is achieved by using the balanced truncation method. Another
global controller, which is more appealing from an implementation point of view, ispresented based on two level PSS to damp both local and interarea modes. Objective
function is presented using GA to allow the selection of the stabilizer parameters. The
effectiveness of the suggested techniques in damping local and interarea modes of
oscillations in multimachine power systems is verified under various operating
conditions to demonstrate their robust performance.
H
Chapter 7: highlights the significant contributions of the present work and draws the
scope for future work in this area.
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Chapter 2
Review of Literature
2.1 Introduction
Power systems are modeled as large nonlinear highly structured systems. Conventional
linear control is limited since it can only deal with small disturbances about an operating
point. Two important issues for power systems control are robustness and a decentralized
structure. The robustness issue arises to deal with sources of uncertainties, which mainly
come from the varying network topology, and the dynamic variation of the load. Since
physical limitation on the system structure makes information transfer among subsystems
unfeasible, decentralized controllers for multimachine systems must be used.
Over the last four decades, considerable amounts of research have been done in the area
of design and application of robust and decentralized control for power system, which are
discussed in the following section.
2.2 Previous Work
The optimal output decentralized (local) and global control of a power system consisting
of three interconnected synchronous machines is considered in [4]. A computational
method is introduced which enables the optimal output control (either global or local) of
a complete system to be found, thereby allowing realistic configurations to be studied.
The main question considered in this paper is as follows: given a multimachine, what are
the advantages, if any of feeding back the outputs of other machines in a system to
control a given machine? How does the performance of such a control system (termed
global control) compare with the overall system performance obtained by controlling
each machine from its own outputs (termed local decentralized control)?
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The load and frequency control of a multi-area interconnected power system is studied in
[5]. In this problem, the system is assumed to be subject to unknown constant
disturbances, and it is desired to obtain, if possible, robust decentralized controllers so
that the frequency and tie-line /net-area power flow of the power system are regulated.
The problem is solved by using some structural results recently obtained in decentralized
control, in conjunction with a parameter optimization method, which minimizes the
dominant eigenvalue of the closed-loop system. A class of minimum order robust
decentralized controllers, which solve this general multi-area load, and frequency control
problem is obtained. Application of these results is then made to solve the load and
frequency control problem for a power system consisting of nine synchronous machines
(described by a 119th-order system).
A method of coordinating multiple adaptive PSS units in a power system is presents in
[6]. The method is based on decentralized adaptive control scheme. Self-tuning adaptive
controllers are used as PSS units on given generators. The generators that tend to strongly
dynamically interact are coordinated by communication the controlled inputs between
them. The communicated information is used in such a way that the controllers are not
dependent on one another and is robust to any communication failures. Simulation results
that compare non-coordinated controllers with coordinated ones are presented for a 17
machines system. Two different system-operating points are tested. It is shown that better
system damping is obtained if the adaptive PSS units on the strongly coupled generators
are coordinated.
The design of a controller for a (TCSC) thyristor controlled series compensator to
enhance the damping of an inter area oscillation in a large power system is presented in
[7]. It describes a comprehensive and systematic way of applying the control design
algorithm in power systems. Two methods to obtain a satisfactory reduced order system
model, which is crucial to the success of the design, are describes.
H
H
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An optimal control method is outlined to deal with uncertainties in power system
modeling and operation as they affect the design of a PSS [8]. It focuses on the design
process for PSS using a nominal model with an uncertainty description, which represents
the possible perturbation of a synchronous generator around its normal operating point.The uncertainties are due to incomplete knowledge of the physical system in the model
formulation process and system abnormal operating conditions. This excitation controller
enables the power system to remain stable over a wide range of operating condition.
H
The design of a robust controller for generator excitation systems is used to improve the
steady state and transient stabilities [9]. The unique approach used is to first treat the
nonlinear characteristics of the system as model uncertainties at the controller design
stage using robust methodology. The performance of the controller has been evaluated
extensively by non-linear simulation. It is concluded that the robust controller provides
better damping to the oscillatory modes of the system than the conventional PSS in all the
cases studied.
In [10], a model matching robustness design procedure based on optimization
theory for the robust redesign of nominal operating conditions and tunes the nominal
control law to enhance the robustness with respect to the off nominal operating
conditions. The procedure is applied to the design of a PSS for a single machine infinite
bus system, which has a range of possible operating conditions. The results show that the
redesign controller contains features similar to the nominal controller, but yet improves
significantly the damping of the machine swing modes at the off nominal conditions.
H
Design of a robust controller for a Static Var Compensator (SVC) to improve the
damping of power system is presented [11]. The main contributions of this paper are to
formulate and to solve the power system damping control problem using robust
optimization techniques, and to synthesize the controller with explicit consideration of
the system operating condition variations. Nonlinear simulations using PSCAD/EMTDC
have been conducted to evaluate the performance of the closed loop system. The results
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have indicated the performance of the closed loop system. The results have indicated that
the designed controller can provide damping to the system under a wide range of
operating conditions.
In [12], the usual trial and error attempts for selection of performance weights are
discarded, that is the main problem in design of a robust
H power system stabilizer
(PSS), and instead a systematic and automated approach based on Genetic Algorithms
(GAs) is proposed. The resulting
H PSS performs quite satisfactory under a wide
range of turbo generator operating conditions and is robust against unmodelled low
damped torsional modes. It also provides sufficient robustness against significant changes
in configuration and parameters.
In [13], a new PSS design for damping power system oscillations focusing on interarea
modes is described. The input to the PSS consists of two signals. The first signal is
mainly to damp the local mode in the area where PSS is located using the generator rotor
speed as an input signal. The second is an additional global signal for damping interarea
modes. Two global signals are suggested; the tie line active power and speed difference
signals. The choice of PSS location, input signals and tuning is based on modal analysis
and frequency response information. These two signals can also be used to enhance
damping of interarea modes using SVC located in the middle of the transmission circuit
connecting the two oscillating groups. The effectiveness and robustness of the new
design are tested on a 19-generator system having characteristics and structure similar to
the Western North American grid.
A method to design sub optimal robust excitation controllers based on control
theory is presented [14]. The sub optimal controller results from additional constraints
that are imposed on the standard optimal
H
H solution. Global stability constraints are
incorporated into the algorithm to ensure stability of the interconnected system
under decentralized control. Furthermore, a lyapunov-based index is used to evaluate the
H
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robustness properties of the closed loop. In order to obtain a reduced order controller, the
method of balanced truncation is used. The sub optimal
H controllers are output
feedback controllers. These controllers posses superior robustness as compared to CPSS
and optimal controllers.H
A method based on optimal control is presented for the design of power system
controllers aimed at damping out electromechanical oscillations [15]. By imposing
decentralization and output feedback as constraints to the control problem, the resulting
controller structures are compatible with those employed by electric utilities. On the other
hand, the use of formulation based on the Chandrasekhar equations allows that sparsity
be exploited by the optimal control algorithm. This makes the method applicable to largesystems. The performance of the proposed method is assessed through its application to
two multimachine systems: the 10 machine New England system and a large power
system based on the South Southeast Brazil interconnected network.
A systematic robust decentralized design procedure based on the optimization
technique for tuning multiple FACTS devices is presented [16]. The design procedure
uses a model matching robustness formulation and requires the design of a parameter toachieve decentralized control. The approach is used to design damping controllers for an
SVC and a TCSC to enhance the damping of the interarea modes in a 3 area 6 machine
system. The feedback signals for the controllers are synthesized from the local voltage
and current measurements.
H
A new method of designing a robust
H PSS to deal with some limitations of the
existing PSS (standard PSSs) is presented [17]. These limitations includeH
H
(i) the inability to treat the system uncertainty when a stable nominal plant becomes an
unstable perturbed plant
(ii) the cancellation of the plants poorly damped poles by the controllers zeros.
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The proposed multiple inputs single output controller for the excitation system is based
on the numerator denominator uncertainty representation which is not restricted in the
modeling of uncertainty as compared to the standard additive or multiplicative
uncertainty representation. Furthermore, the bilinear transformation has been used in the
design to prevent the pole zero cancellation of the poorly damped poles and to improve
the control system performance. Simulation results have shown satisfactory performance
of this PSS for a wide range of operating conditions and good stability margin as
compared to both the conventional PSS and the standard
H PSS.
The decentralized load frequency controller design problem presented [18]. It is shown
that, subject to a condition based on the structure singular values (SSV), each local area
load frequency controller can be designed independently. The stability condition for the
overall system can be stated as to achieve a sufficient interaction margin and a sufficient
gain and phase margin defined in classical feedback theory during each independent
design. It is demonstrated by computer simulation that within this general framework,
very local controllers can be designed to achieve satisfactory performances for a sample
two-area power system and a simplified four-area power system. Under the designed
framework based on the structure singular values, other design methods for local area
controllers may be applied.
In [19], the design of linear robust decentralized fixed structure power system damping
controllers using GA is presented. The designed controllers follow a classical structure
consisting of a gain, wash out stage and two lead lag stages. To each controller is
associated a set of three parameters representing the controller gain and the controller
phase characteristics. The GA searches for an optimum solution over the parameter
space. Controller robustness is taken into account as the design procedure considers a
prespecified set of operating conditions to be either stabilized or improved in the sense of
damping ratio enhancement. A truly decentralized control design is achieved as the loop
control channels are closed simultaneously. The approach is used to design SVC and
TCSC damping controllers to enhance the damping of the interarea modes in a three-area
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six-machine system. Local voltage and current measurements are used to synthesis
remote feedback signals.
A new decentralized nonlinear voltage controller for multimachine power systems is
introduced [20]. A decentralized nonlinear voltage controller is developed by use of the
robust control theory. Performance of this controller in a three-machine example system
is simulated. The simulation results show that both voltage regulation and system
stability enhancement can be achieved with this controller regardless of the system
operating conditions.
In [21], coordinated optimal decentralized controller design of excitation and TCSC
control for improving damping of overall power systems is discussed. Decentralized and
coordinated control is indispensable to power system because power systems are large
scaled and geographically distributed over large area. In particular, FACTS devices need
decentralized and coordinated control more and more. One present the decentralized
controller based on the only local available output variable of each subsystem. Simulation
of 3 machine and 9 bus with 1 TCSC show that decentralized controller has the
reasonable performance compared to centralized controller.
A novel method for the design of TCSCs in a meshed power system is developed [22].
The selection of the output feedback gains for the TCSC controllers is formulated as an
optimization problem and the simulated annealing (SA) algorithm is used to find the
solution. Using this method, the conflicting design objectives, such as the improvement
in the damping of the critical modes, any deterioration of the damping of the non-critical
modes and the saturation of the controller actuators, can be simultaneously considered. It
is also shown that the SA algorithm can be used to design robust controllers, which
satisfy the required performance criteria over several operating conditions. This control
scheme can be easily implemented as only the measurable signals local to each TCSC
location are used to control the TCSCs (decentralized control).
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In [23], one presents a new setup for analyzing the stability of a multimachine power
system with parameter variations. This method is based on SSV, which allows
computation of an effective measure for robustness in the presence of real parametric
uncertainty. Once the robustness problem has been set up it becomes amenable to the
application of synthesis tools for robust controller design. This technique is applied to
the robust stability assessment of a 4-machine test system specifically designed to
analyze the effect of control on the interarea.
In [24], numerical simulations and testing results of the new method is presented for
stability robustness of multimachine power systems. The approach is based on SSV tools.
The variations of operating conditions are treated as structured uncertainty. Simulation
results for a test system have shown excellent accuracy of robust stability assessment for
a wide range of operating conditions.
In [25], a systematic procedure for the design of decentralized controllers for
multimachine power systems is presented. The robust performance in terms of (SSV or
) is used as the measure of control performance. A wide range of operating conditions
was used for testing. Simulation results have shown that the resulting controllers
would effectively enhance the damping torques. Providing better robust stability and/or
performance characteristics both in the frequency and time domain compared to
conventionally designed PSSs.
A robust decentralized excitation control of multimachine power systems is introduced
[26]. One concerned with the design of decentralized state feedback controller for the
power system to enhance its transient stability and ensure a guaranteed level of
performance when there exist variations of generator parameters due to changing load
and/or network topology. It is shown that the power system can be modeled as a class of
interconnected systems with uncertain parameters and interconnections. One develops a
guaranteed cost control technique for the interconnected system using a linear matrix
inequality ( LMI ) approach. A procedure is given for the minimization of the cost by
employing the powerful LMI tool. The designed controller design is simulated for a
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three-machine power system example. Simulation results show that the decentralized
guaranteed cost control greatly enhances the transient stability of the power system in the
face of various operating points, faults in different locations or changing network
parameters.
Robust design of multimachine PSSs using SA optimization technique is presented [27].
This approach employs SA to search for optimal parameter settings of a widely used
conventional fixed structure lead lag PSS (CPSS). The parameters of this simulated
annealing based power system stabilizer (SAPSS) are optimized in order to shift the
system electromechanical modes at different loading conditions and system
configurations simultaneously to the left in the s-plane. Incorporation of SA as a
derivative free optimization technique in PSS design significantly reduces the
computational burden. One of the main advantages of this approach is its robustness to
the initial parameter settings. In addition, the quality of the optimal solution does not rely
on the initial guess. The performance of the SAPSS under different disturbances and
loading conditions is investigated for two multimachine power systems. The eigenvalue
analysis and the nonlinear simulation results show the effectiveness of the SAPSSs to
damp out the local as well as the interarea modes and enhance greatly the system stability
over a wide range of loading conditions and system configurations.
Robust design of multimachine (PSSs) using the Tabu Search (TS) optimization
technique is presented [28]. This approach employs TS for optimal parameter settings of
a widely used conventional fixed structure lead lag PSS (CPSS). The parameters of this
stabilizer are selected using TS in order to shift the system poorly damped
electromechanical modes at several loading conditions and system configurations
simultaneously to a prescribed zone in the left hand side of the s-plane. Incorporation of
TS as a derivative free optimization technique in PSS design significantly reduces the
computational burden. In addition, the quality of the optimal solution does not rely on the
initial guess. The performance of this PSS under different disturbances and loading
conditions is investigated for multimachine power systems. The eigenvalue analysis and
the nonlinear simulation results show the effectiveness of the designed PSSs in damping
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out the local as well as the interarea modes and enhance greatly the system stability over
a wide range of loading conditions and system configurations.
Several control design techniques namely, the classical phase compensation approach, the
synthesis, and a linear matrix inequality technique, are used to coordinate two power
system stabilizers to stabilize a 5-machine equivalent of the South/Southeast Brazilian
system [29]. The open loop system has an unstable interarea mode and cannot be
stabilized using only one conventional power system stabilizer. Both centralized and
decentralized controllers are considered. The different designs are compared and several
interesting observations are provided.
An effective method for designing coordinated
H PSSs to improve the damping of
local and interarea oscillations is presented [30]. Target modes types of inputs of the PSS,
and effective locations for each controller are examined using the participation factor and
residue concept. To realize coordination of the controllers, a method for constructing the
effective reduced model for this design is presented, minimizing the uncertainty for each
controller. With such a small uncertainty, a tight design, which yields marginal
robustness, can be realized, increasing the performance of each controller as well as that
of the total system. The influence of the reduced model on the controller characteristics is
discussed. The effectiveness of this design is demonstrated through nonlinear numerical
simulation in a five machine seven bus system under two critical operating conditions.
A design method of damping controllers of two facts devices, namely synchronous
voltage source when it is used only for reactive shunt compensation, advanced static var
compensator (ASVC) and SVC is introduced [31]. The application of ASVC and SVC for
damping control is demonstrated and the comparison is made about the damping controlcapabilities paying attention to the difference in the design philosophy and detailed
dynamic performances. An important issue in designing this kind of controllers is to
suppress over voltage that appears under large disturbance. This over voltage problem
sometimes appears in the existing SVC system. To cope with this over voltage problem,
it uses the control sensitivity function to regulate indirectly the controller output so that
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the over voltage problem is treated in the design step. Another important issue is that one
point out the zero problems tends to appear inherent in a typical ASVC system making
the controller design difficult. To design robust controller under this condition, one
suggest to use the bilinear transform to design a robust
H optimal controller, it is
shown that this controller provides more robust stability and better performance for
additional damping for power system oscillations while suppressing over voltages.
Performance comparison is also made between ASVC and SVC cases.
A new PSS design method, which uses the numerator denominator perturbation
representation and includes the partial pole placement technique and a new weighting
function selection method is presented [32]. This overcomes certain conventionalPSS design algorithm limitations. A sixth order machine model is used to increase
the accuracy of selected weighting functions. A robust PSS has been successfully
designed for single and two machine systems by treating the highly nonlinear
characteristic of the power system as model uncertainty. The design is verified to have
better performance for a wide range of operating conditions when compared with the
conventional PSS designs.
H
H
The design of robust power system stabilizers, which place the system poles in an
acceptable region in the complex plane for a given set of operating and system
conditions, is introduced [33]. It therefore, guarantees a well-damped system response
over the entire set of operating conditions. The proposed controller uses full state
feedback. The feedback gain matrix is obtained as the solution of a LMI expressing the
pole region constraints for polytopic plants. The technique is illustrated with applications
to the design of stabilizers for a single machine and a 9 bus, 3 machine power system.
The design of robust control for the second generation of FACTS devices such as static
compensator (STATCOM), and unified power flow controller (UPFC) using a loop
shaping design via a normalized coprime factorization approach, where loop shape refers
to the magnitude of the loop transfer function L=GK as a function of frequency is
H
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presented [34]. Since method is based on classical loop shaping ideas, it is relatively easy
to implement. Furthermore, comparing it with the other methods of robust control,
it is more flexible and is not limited in its applications. Simulation of the system
following a disturbance is performed to demonstrate the effectiveness of the designedcontroller.
H
A robust controller for providing damping to power system transients through
STATCOM devices is presented [35]. The method of multiplicative uncertainty has been
employed to model the variations of the operating points in the system. A loop shaping
method has been employed to select a suitable open loop transfer function, from which
the robust controller is constructed. The design is carried out applying robustness criteria
for stability and performance. The proposed controller has been tested through a number
of disturbances including three phase faults. The robust controller designed has been
demonstrated to provide extremely good damping characteristics over a range of
operating conditions.
A genetic algorithm based method is used to tune the parameters of a PSS [36]. This
method integrates the classical parameter optimization approach, involving the solution
of a Lyapunov equation, within a genetic search process. It also ensures that for any
operating condition within a predefined domain, the system remains stable when
subjected to small perturbations. The optimization criterion employs a quadratic
performance index that measures the quality of system dynamic response with in the
tuning process. The solution thus obtained is globally optimal and robust. This method
has been tested on two different PSS structures: the lead lag PSS and the derivative PSS.
System dynamic performance with PSS tuned using this technique is highly satisfactory
for different load conditions and system configurations.
A mixed sensitivity design of a damping device employing a UPFC is presented
[37]. The problem is posed in the LMI framework. The controller design is aimed at
providing adequate damping to interarea oscillations over a range of operating conditions.
H
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The results obtained in a two area four machine test system are seen to be very
satisfactory both in the frequency domain and through nonlinear simulations.
In [38], a robust PSS is designed using loop shaping design procedure. The resulting PSS
ensures the stability of a set of perturbed plants with respect to the nominal system and
has good oscillation damping ability. Comparisons are made between the resulting PSS, a
conventionally designed PSS and a controller designed based on the structure singular
value theory.
A decentralized controller for load frequency control (LFC) problem in power systems is
designed based on control technique formulated as a LMI problem [39]. To
achieve decentralization, interfaces between interconnected power systems control area
are treated as disturbances. The LMI control toolbox is used to solve such a constrained
optimization problem for LFC applications. The performance of this controller is
illustrated and compared with that of a conventional controller through simulation of a
two-area power system.
H
A LMI based robust controller design for damping oscillations in power systems is
presented in [40]. This controller uses full state feedback. The feedback gain matrix is
obtained as the solution of a LMI. The technique is illustrated with applications to the
design of stabilizer for a typical single machine infinite bus (SMIB) and a multimachine
power system. The LMI based control ensures adequate damping for widely varying
system operating conditions and is compared with conventional power system stabilizer
(CPSS).
2H
In [41], a design procedure of a H mixed sensitivity PSS is developed to improve
power system stability. A study system representing SMIB is investigated. The machine
accelerating torque is selected as input signal to the PSS. A comparison between system
response to disturbances for the
H and lead lag PSS is made. The simulation results
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show that the PSS ensures the stability of the system and has good damping ability
at a wide range of system loading.
H
The Particle Swarm Optimization (PSO) technique is used to develop a controller fordamping power system oscillations [42]. The speed deviation and its rate of change are
selected as input signals to the controller. The objective is to get optimal gains values of
the controller within pre-specified limits to improve the system dynamics. In order to
ensure the reliability of the PSO based controller, a comparison has made between the
effect of the developed controller and that of
H controller on the dynamic
performance of a SMIB. The simulation results show that the PSO based controller offers
effective damping to system oscillations in a wide range of operating conditions.
The application of PSO technique to optimize a PID controller parameters for LFC is
discussed [43]. The capability of the controller is investigated through variations the
magnitude of load disturbance. The simulation results show that the applied PSO based
PID controller has achieved good system performance. A comparative study results is
made between the controller and the designed one. The performance is shown to be
better for the new PID controller.
H
2.3 Contributions of This Thesis
There has been considerable amount of work done to develop new controllers in the area
of design and application of robust and decentralized control for power system. The
previous controllers are suffered from high dimension and practical implementation
especially in multimachine system. Moreover, none of them addresses the problem of
coordination of multiple controllers by means of global controller. This research has
filled this point since it has achieved the design, development, and testing of a simple low
order controller and global controller scheme to control and coordinate the actions of
decentralized controllers. The proposed controller provides a simple and effective scheme
to stretch the stability limit and to increase the loadability of the system.
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Chapter 2 Review of literature
The main contributions of this thesis are summarized as follows:
A new controller is presented to design a robust PSS for a SMIB using PI controller. The
parameters of PI controller are obtained by GA to achieve the same performance of
H
based on output feedback in term of LMI. This controller, which is called GALMI, is
simpler than one. This controller succeeds in achieving a robust tuned PSS.
Moreover, it overcomes the difficulty of computation and high dimension of controller
system especially in multimachine environment. So it is more advantageous, in terms of
practicability and reliability.
H
The design of centralized and decentralized PSS for a multimachine power system usingoutput feedback is presented. In centralized controller, the control signal is a
function of output of all machines. In decentralized controller, the control signal to each
machine becomes a function of the output of that machine only.
H
A new simple algorithm is introduced to design robust decentralized PSS for a
multimachine power system using GALMIs. For each area, the decentralized controller
based on is replaced by GALMI. Moreover, a global controller is designed to deal
with the interactions, which is unconsidered in design of decentralized controllers.
H
A new application for a MIMO centralized
H controller is illustrated for designing a
high order global controller. Another two global controllers are introduced in this thesis
to treat the problem of possible adverse interaction between multiple decentralized
controllers and to overcome the problem of high order of centralized controller.
The first is based on the reduced centralized with minimum communicated
information. While the second is based on two level PSS. In this controller, not only is
the cost of implementation drastically reduced, but also, the risk of loss of stability due to
signal transmission failure is minimized. Moreover, it represents a simple global
controller to damp both local and interarea modes.
H
H
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Chapter 3 Modeling of Power Systems
Chapter 3
Modeling of Power Systems
3.1 Introduction
The nonlinear differential equations governing the behavior of a power system can be
linearized about a particular operating point, to obtain a linear model, which
represents the small signal oscillatory response of a power system. Variations in the
operating condition of the system result in the variations in the parameters of the
small signal model. A given range of variations in the operating conditions of a
particular system thus generates a set of linear models, each corresponding to one
particular operating condition. Since, at any given instant, the actual plant could
correspond to any model in this set, a robust controller would have to impart adequate
damping to each one of these entire sets of linear models. In this chapter, the
mathematical models for a power system required in formulating the stability problem
will be presented. The typical single machine infinite bus (SMIB) is shown in Figure
(3.1). The system data has been given in appendix A
Figure (3.1) Machine-infinite bus system
Xe Re
3.2 System Equations
The complete system has been simulated in state space representation. For simplicity
both the state and algebraic equations for (SMIB) are given below [1,2]:
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Chapter 3 Modeling of Power Systems
Machine Equations
Using the third order model for the synchronous machine, the differential equations,
which describe the machine dynamics, can be arranged as
dI
do
dX
dX
fdE
doq
E
doq
E
+
+
=
11& (3.1)
eT
mT
j
=1
& (3.2)
= B&
(3.3)
Static Exciter Equations
The differential equations that describe the static excitation system can be written as
sVaa
K
tVaa
K
fVaa
K
fdEa
fdE +
= 1
& (3.4)
sV
fa
fKa
K
tV
fa
fKa
K
fV
fa
fKa
K
ffd
E
fa
fK
fV +
+
=
1& (3.5)
Algebraic Equations
The linearized algebraic equations can be summarized as
qI
qX
dV = (3.6)
dI
dX
qE
qV += (3.7)
dI
qoI
dX
qX
qI
qoE
qE
qoI
eT
+= (3.8)
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Chapter 3 Modeling of Power Systems
qV
toV
qoV
dV
toV
doV
tV += (3.9)
The transmission network having an impedance ofe
jXe
R + , which is connected to
an infinite bus with voltage , is included in the following equations:
V
dI
eRV
qI
eX
dV =
+ ])
0cos([ (3.10)
qI
eRV
dI
eX
qV =
+ ])
0sin([ (3.11)
3.3 Block Diagram Simulation
One has to build the block diagram of every equation and all blocks are connected
together to form one block. For example, the machine equation. (3.1) is formed as
(Xd-X'd)I
d
Efd
+
+
1
1
+ sdo
E'q
The exciter is connected to this block by the voltagefd
E , i.e., the output of the
exciter is an input to the field winding of the machine.
+
+
-
-
Vref
Vt
Vs
EfdKa
(1+ as)
(1+ fS)
KfS
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Chapter 3 Modeling of Power Systems
The terminal voltage Vt that is a feedback signal to the exciter is formed by the
following block.
Iq
Id
E'q
-XqVd
VqX'd +
+
Vdo Vto
VqoVto
+
+
Vt
The electric torque equation (3.8) is formed in a block diagram as
Iq
Id
E'q
Eqo
X'd)(Xq- Iqo
Iqo
Te
+
+
-
The electric torque output signal from the previous block is an input to the block
diagram representing the swing equation, which is described by equation (3.2) and
(3.3).
Tm
Te
+
-
1
jS
o
S
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Chapter 3 Modeling of Power Systems
The transmission network equation (3.10,3.11) are formed in a block diagram as
shown below
V cos(0-)
Xe1/Re
+
+
Iq
Vd
Id
V sin(0-)
Xe1/Re
+
+
+
Id
Vq
Iq
The block diagram of a single machine adopted to be used in SIMULINK Toolbox is
developed as shown in Figure (3.2).
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Chapter 3 Modeling of Power Systems
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Chapter 3 Modeling of Power Systems
3.4 State Space Formulation
The system in general is described by a set of differential and algebraic equations in
the standard form as shown in equation (3.12)
(3.12)]][[]][[][
]][[]][[][
UDXCY
UBXAX
+=
+=&
Once the equations are obtained in this form the eigenvalues of the matrix A indicate
the stability of the system.
In general these equations can be written as shown previously but they can be
rearranged as follows:
(3.13)[ ] [ ][ ] [ ][ ]URXQY
XP +=
&
Where X, Y and U are the state space variables, while P, Q and R are real constant
matrices. The entries of these matrices are function of all system parameters and
depend on the operating condition. The P matrix of equation (3.13) can be partitioned
as follows:
[ ] (3.14)
=
xsG
oAI
P0
Where matrix I is an identity matrix of dimension , where is the number of
the state space variables. The P matrix is of dimension n x n where n is the total
number of states and algebraic variables. Matrix 0 is a null matrix. The matrix
snx
sn
sn
sG is
a square matrix of dimension wherev
nxv
nv
n is the total number of the algebraic
variables and matrixA0 is a very sparse matrix of dimension vnxns . Then the inverse
of P matrix is obtained by partitioning as follows:
(3.15)[ ]
=
1][0
1][1
xsG
xsG
oAI
P
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Chapter 3 Modeling of Power Systems
Thus the inverse of P matrix involves the inversion of the sG matrix, which ispartitioned in turn, and it is almost in a block form.
The Q and R matrices are partitioned as
=
=
DR
sR
R
CQA
QQ ,
Where is a matrix of dimension and matrixA
Qs
nxs
n C
Q is of dimensions
nxv
n
then the coefficient matrices of the state space are
DR
sGD,DoA
sRB
Co
AA
QA,c
Qxs
GC
1][
1][
==
=
=
It can be seen that the matrixAis obtained, as a sum of two matrices. The matrix
contains almost all the control parameters. The eigenvalues of the system matrix A
described by the equation (3.12) are indicative of the system performance.
A
Q
The system eigenvalues are related to the different modes in the system while the real
part is a measure of the amount of the damping and the imaginary part is related to the
natural frequency of the oscillation of the corresponding modes. System eigenvalues
are in general function of all control and design parameters; the change in any of these
parameters affects on the system performance. Hence, causes a shift in the whole
eigenvalue pattern. The amount of shift depends on the sensitivity of the different
eigenvalues as well as the amount of change in the parameter. The matrix P is shown
below.
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Chapter 3 Modeling of Power Systems
=
eR
eX
eX
eR
tV
qV
tV
dV
qEqXqXqI
dX
qX
fT
aT
fK
aK
aT
aK
j
d
dX
dX
P
001000000
000100000
0010
0
0
0
000000
00010000000
0001000000
0000100000
0000010000
0000001000
00000000100
0001
0000010
0
0
000000001
Equation (3.13) can be written in the form
[ ]sfTaTf
K
a
Ka
T
aK
fVfd
E
qE
V
V
qI
fT
aT
fKa
K
fT
fT
aT
fK
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